A290789 A(n,k) is the n-th Carlitz-Riordan q-Catalan number (recurrence version) for q = -k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -1, 1, 1, 1, -2, -7, 0, 1, 1, 1, -3, -23, 47, 2, 1, 1, 1, -4, -55, 586, 873, 0, 1, 1, 1, -5, -109, 3429, 48778, -26433, -5, 1, 1, 1, -6, -191, 13436, 885137, -11759396, -1749159, 0, 1, 1, 1, -7, -307, 40915, 8425506, -904638963, -8596478231, 220526159, 14, 1
Offset: 0
Examples
Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, ... 1, 0, -1, -2, -3, -4, ... 1, -1, -7, -23, -55, -109, ... 1, 0, 47, 586, 3429, 13436, ... 1, 2, 873, 48778, 885137, 8425506, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..55, flattened
- J. Fürlinger, J. Hofbauer, q-Catalan numbers, Journal of Combinatorial Theory, Series A, Volume 40, Issue 2, November 1985, Pages 248-264.
- Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Masterarbeit, University of Vienna. Fakultät für Mathematik, 2013.
Crossrefs
Programs
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Maple
A:= proc(n, k) option remember; `if`(n=0, 1, add( A(j, k)*A(n-j-1, k)*(-k)^j, j=0..n-1)) end: seq(seq(A(n, d-n), n=0..d), d=0..12); -
Mathematica
Unprotect[Power]; Power[0|0., 0|0.]=1; Protect[Power];A[n_, k_]:=A[n, k]=If[n==0 , 1, Sum[A[j, k] A[n - j - 1, k]* (-k)^j, {j, 0, n - 1}]]; Table[A[n, d - n], {d, 0, 15}, {n, 0, d}] (* Indranil Ghosh, Aug 13 2017 *) -
Python
from sympy.core.cache import cacheit @cacheit def A(n, k): return 1 if n==0 else sum(A(j, k)*A(n - j - 1, k)*(-k)**j for j in range(n)) for d in range(16): print([A(n, d - n) for n in range(d + 1)]) # Indranil Ghosh, Aug 13 2017
Formula
G.f. of column k: 1/(1-x/(1+k*x/(1-k^2*x/(1+k^3*x/(1-k^4*x/(1+ ... )))))).
A(n,k) = Sum_{j=0..n-1} A(j,k)*A(n-j-1,k)*(-k)^j for n>0, A(0,k) = 1.